%%
%% functionalScalar.tex
%% 
%% Made by Alex Nelson
%% Login   <alex@tomato>
%% 
%% Started on  Sat Aug 15 13:06:34 2009 Alex Nelson
%% Last update Sat Aug 15 13:06:34 2009 Alex Nelson
%%

We would like to generalize our functional methods to field
theory by generalizing the parametrization from a 1 dimensional
parameter (time $t$) to a 4 dimensional parametrization
$x^{\mu}$. So this means that
\begin{align*}
t&\to x^{\mu}\\
\frac{d}{dt}&\to \partial_{\mu}\\
q(t)&\to \varphi(x^{\mu})
\end{align*}
In fact we can set up a dictionary for our generalization
\begin{center}
\begin{tabular}{|c|c|}\hline
mechanics          & field theory\\\hline
$q(t)$             & $\varphi(\bar{x},t)$\qquad{classical field}\\
$\widehat{Q}$      & $\varphi(\bar{x},t)$\qquad{operator field}\\
$f(t)$             & $J(\bar{x},t)$\qquad{classical source}\\\hline
\end{tabular}
\end{center}
We can write the Hamiltonian density for the free scalar field as
\begin{equation}%\label{eq:}
\mathcal{H}_{0} = 
\underbracket[0.25pt]{\frac{1}{2}\Pi^{2} +
\frac{1}{2}(\nabla\varphi)^{2}}_{\mathclap{\substack{=\partial^{\mu}\varphi\partial_{\mu}\varphi\\
=\text{kinetic energy}}}}+
\underbracket[0.25pt]{\frac{1}{2}m^{2}\varphi^{2}}_{\mathclap{\text{potential}}}
\end{equation}
The trick we've employed so many times before using
infinitesimals is done precisely the same way using the
Hamiltonian density. That is,
$\mathcal{H}_{0}\mapsto(1-i\varepsilon)\mathcal{H}_{0}$. For
simplicity, we tacitly assume we always mean
$(1-i\varepsilon)m^{2}$ whenever we write $m^{2}$, it's
completely equivalent to
$\mathcal{H}_{0}\mapsto(1-i\varepsilon)\mathcal{H}_{0}$. 

Consider the functional integral
\begin{equation}%\label{eq:}
Z[J] = \<0|0\>_{J} = \int\mathcal{D}\varphi\exp\left[i\int[\mathcal{L}_{0}+J\varphi]d^{4}x\right]
\end{equation}
where
\begin{equation}%\label{eq:}
\mathcal{L}_{0} =
\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi - \frac{1}{2}m^{2}\varphi^{2}
\end{equation}
is the Lagrangian density and
\begin{equation}%\label{eq:}
\mathcal{D}\varphi\propto
\prod_{\mathclap{\substack{
\text{space-}\\
\text{time}\\
\text{points $x$}}}}
d\varphi(x)
\end{equation}
is the functional measure.

\subsubsection{Remark on Peskin and Schroeder's ``Derivation''.}
In Chapter 9 of Peskin and Schroeder~\cite{Peskin:1995ev} there
is a ``derivation'' of the functional quantization of the scalar
field using the limit of a discrete lattice. There is some
mathematically dubious aspects of this derivation, for one they
use a Fourier series when they should be using a discrete Fourier
transform. What difference does this make? Well, the intuition of
the discrete Fourier transform is that the field exists only on
the nodes of the lattice; the intuition of a Fourier series is
that it exists everywhere continuously in a finite universe. The
latter is not in spirit with taking the continuum limit! In fact,
this beautifully motivates us to read the paper by Wise on chain field
theory~\cite{Wise:2005mp}, which resolves this problem by a
topological approach which yields the correct continuum limit by
taking advantage of recent work on discrete differential geometry.

\subsection{Slick Method of Computing Correlation Functions}
\input{tex/functionalScalarFieldSlick}
